OF TURBINATED AND DISCOID SHELLS. 361 



simply as the linear dimensions of the animal or its shell, offers an analogy, and has 

 perhaps a relation, to the increase of the section of the shell, according to the same 

 law of its simple linear dimensions. 



Subjoined to this paper is a mathematical discussion of the following geometrical 

 and mechanical elements of a conchoidal surface : its volume, the dimensions of its 

 SURFACE, the CENTRE OF GRAVITY of its Contained solid, the centre of gravity of its 

 surface. 



These elements are determined (the law of the logarithmic spiral being supposed) 

 by certain transcendental functions, having constant factors dependent for their 

 amount upon the statical moments and the moments of inertia of the generating 

 figures and of their areas. 



The object proposed in the determination of these elements was their application 

 to a discussion of the hydraulic theory of shells ; yet further, if'possible, to develope 

 that wisdom of God which shaped them out and moulded them ; and especially in 

 reference to the particular value of the constant angle which the spiral of each spe- 

 cies of shell affects, — a value connected by a necessary relation with the economy of 

 the material of each, and with its stability, and the conditions of its buoyancy*. 



The paper concludes with a discussion of the general equations to a conchoidal 

 surface in respect to systems of polar and of rectangular coordinates. 



To determine the Volume of a Conchoidal Solid. 



Suppose the generating curve to be a plane curve, and let it (first) retain its position 

 upon the axis as it revolves, varying its dimensions. 



Let P C and Q C (fig. 3.) be two of its positions, inclined at the angle A 0, and in- 

 cluding between them the elementary solid P C Q. 



Imagine the plane P C to have revolved about A z through the angle A without 

 altering its dimensions, the solid generated by it would then, by the theorem of Gul- 

 DiNUs, be represented by M . A 0, where M represents the statical moment of the 

 plane P C about the axis A z. 



The elementary solid imagined to be in like manner generated by the revolution 

 of Q C through the angle A 0, will similarly be represented by (M -{- A M) A 0. 



Now between these two imaginary solids is evidently the actual elementary solid 

 P C Q. Calling then V the volume of the solid to be determined, we have 



M A < A V < (M -I- A M) A 0. 



Or, considering M and V as functions of 0, and expanding by Taylor's theorem, 



MA0<|^A© + ^,-i^ + &c.<MAe + |^(A0)2 + &c. 



* As illustrative of this remark, it may here be mentioned that the shell of the Nautilus Pompilius has, hy- 

 drostatically, an ^-statical surface. If placed with any portion of its surface upon the water, it will imme- 

 diately turn over towards its smaller end, and rest only on its mouth. Those conversant with the theory of 

 floating bodies will recognise in this an interesting propert)'. 

 MDCCCXXXVIII. 3 A 



